We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. In this paper we describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth �number of edges from the root) of the most recent common ancestor of a random subset of k species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs

In testing the independence of two Gaussian populations, one computes the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The “Laplace transform” of this distribution is not only an integral over the Grassmannian of p-dimensional planes in complex n-space, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations. They also have expansions, related to random words of length ℓ formed with an alphabet of p letters. Given that each letter appears in the word, the maximal length of the disjoint union of p increasing subsequences of the word clearly equals ℓ. But the maximal length of the disjoint union of p − 1 increasing subsequences leads to a non-trivial distribution. It is precisely this

All graphs considered in the paper are directed. Let be a graph on n vertices which we identify with the elements of the additive cyclic group Z n f0; 1;...;n 1g. The graph is called circulant if it has a cyclic symmetry, that is, if the permutation ð0; 1; 2;...;n 1Þ is anautomorphism of the graph.

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We discuss the problem to count, or, more modestly, to estimate the number f(m, n) of unimodular triangulations of the planar grid of size m n.

Abstract. In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings.

Abstract. Local Weyl modules over two-dimensional currents with values in glr are deformed into spaces with bases related to parking functions. Using this construction we 1) reproof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; 2) describe the limit of Weyl modules in terms of semi-infinite forms and find the limit of characters; 3) propose a lower bound and state a conjecture for dimensions of Weyl modules with arbitrary highest weights. Also we express characters of deformed Weyl modules in terms of ρ-parking functions and the Frobenius characteristic map.

We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions. PACS number(s): 05.50.+q, 02.10Ab, 61.41.+e

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